Partially interconnected topological network

ABSTRACT

In a partially interconnected topological network having at least six topological nodes, a topological node being a single physical node or a group of interconnected physical nodes or part of a physical node or a group of interconnected physical nodes and parts of physical nodes, each topological node having at least three point-to-point topological links connecting it to some, but not all, of the plurality of topological nodes, there is at least one choice of routing between any two topological nodes, where the choice of routing is either two point-to-point topological links connected in series at another of the topological nodes or a direct point-to-point topological link between the two topological nodes.

BACKGROUND OF THE INVENTION

The role of network design in providing Quality of Service (QoS) isoften overlooked: a poorly designed network architecture will never,regardless of the sophistication of the bandwidth allocation strategyused, be able to match the performance of a well designed network. Belowis shown a type of network that overcomes many of the problemsassociated with conventional network designs. These networks havebounded hop counts, relatively few links and an even distribution ofroutes across the network. Even route distribution allows path selection(i.e. routing) algorithms to evenly load traffic onto the network,preventing network hotspots that degrade performance; an even loaddistribution improves the networks response to failure. Consequently,these regular partially-meshed networks have exceptional performance.Firstly, existing strategies for designing networks, and their problems,are analysed.

Problems of Network Design

Communications networks are very complex systems. They consist of manyphysical and logical layers, built from many different technologies. Anetwork can be viewed as wholly physical (e.g. a Synchronous DigitalHierarchy (SDH) transport network built of duct, fibre and switchingequipment), part physical, part logical (e.g. a set of AsynchronousTransfer Mode (ATM) or Internet Protocol (IP) switches built fromlogical links provided by a transmission/transport network) or whollylogical (e.g. the logical nodes and links of an ATM PrivateNetwork-Network Interface (PNNI) hierarchy). In this many-layeredstructure, each layer will need a different network design, (whichshould ideally take into account higher and lower level layers), toaccount for differing design objectives, constraints and technologicallimitations. The use of various transmission arrangements is possiblewhen carrying a point-to-point topological link. For example such a linkcan be the whole or just part of a transmission system, severaltransmission systems in series, or a combination of such arrangements. Apoint-to-point topological link may also itself carry many circuits orpacket streams which are multiplexed together.

In any real network traffic must enter and exit the network at varioustopological nodes. This should be assumed to apply for the variousconfigurations, although it may not be specifically mentioned.

There may be cases where some constraint imposes that a combination oftopological structures may be combined to form a larger network, inwhich case only part of an overall network may consist of a regularpartially-meshed network. Similarly, an overall network may contain morethan one instance of a regular partially-meshed network.

Concentration is on the topological aspects of network design and notfocussed on any particular implementation technology, though it isanticipated that the designs will find more application at the servicelayers of a network (IP/ATM/Public Switched Telephone Network (PSTN))rather than at the transport layer. The network is therefore designed asa topological arrangement of nodes and links that provides connectivitybetween all pairs of nodes.

Ring and mesh networks are a good starting point for a discussion ofnetwork design, since they illustrate the trade-offs inherent in allnetworks. Suppose a network is required to carry a constant amount oftraffic C between all pairs N of nodes. If every node is connected toevery other node, the result is a fully-meshed network as shown in FIG.1 a, which has of order N² links (i.e. N(N−1)/2 links). Each of theN(N−1) streams of traffic is switched twice, once at the originatingnode, and once at the destination node. Each stream of traffic has onehop between source and destination (where a hop is defined as the numberof links traversed). An alternative is to connect all nodes into a ringusing N links as shown in FIG. 1 b. Traffic between non-adjacent nodeshas to be switched by intervening nodes and the average number of hopsper traffic stream is proportional to N. More traffic carrying capacityhas to be provided by the nodes and links to carry this transit traffic.The total transit traffic in a ring network scales as N³, for N even,the transit traffic is CN(N/2−1)², for N odd it isCN((N+1)/2−1)((N+1)/2−2)

Which of these two is better? The fully-meshed network makes efficientuse of its nodes: no capacity is used switching transit traffic.However, too many links are used to achieve this objective; the numberof links grows so fast with N, that meshes are only practical for smallnetworks. If network ports are a limited resource, then the maximumnumber of nodes in a network (and hence its capacity) is limited by thenumber of ports. Is the ring network any better? In a ring network,network size is not limited by the number of ports, but, as the size ofthe network increases, more and more capacity has to be devoted tocarrying transit traffic. In both cases the designs do not scale well: aring has too many hops; a filly-meshed network has too many links.

Networks with better scaling properties can be designed by adding linksselectively between nodes to form a ‘Random Partial Mesh’ as shown inFIG. 2 with nine nodes. Typically each node will be connected to atleast two other nodes to ensure the network survives a link failure, andit is ensured that no single node failure can split the network into twopieces. As we shall see, a random partial-mesh represents, at least insome respects, a good compromise between ring and mesh networks. Inparticular performance and cost can be traded by altering the degree ofmeshing.

Designing networks from scratch to meet specific objectives, such ascost and performance is intrinsically difficult: the optimisationproblem is NP-complete. That is, no algorithm exists to find an optimumsolution in polynomial time. Many techniques, such as simulatedannealing or genetic algorithms, can be used to find sub-optimal, butuseful, solutions, and some design tools incorporate these algorithms orother heuristics. However, the major problem with these approaches isthat the quality of the design can only ever be as good as theoptimisation criteria used. Choosing practical optimisation criteria isin itself a difficult problem. Some constraints may find expression in acost function, but adequate solutions may not be found because ofinadequacies in the search algorithms.

Analysing the performance of a randomly-meshed network is difficult dueto its irregularity. It is a problem best tackled by a computer.Analysis of the routes in a randomly-meshed network (which routes are afunction of the topology of the network) almost always shows that somenodes act as ‘hubs’, concentrating many short routes (of length two orthree hops). In consequence, the hub nodes and surrounding links will,when loaded using almost any routing protocol, become more highlyutilised, since every routing protocol will utilise shorter routesfirst. In other words the network will develop hot spots, not as aresult of asymmetric traffic, but because of the network topology. Thisneed not represent a problem, provided the network is dimensioned tosupport the asymmetric traffic. However, the consequences of node orlink failure can be serious, particularly when a hub node or attachedlink fails.

Regular Partially-Meshed Networks

Scalability and hub problems can be overcome if partially meshed,regular networks (Regular Partial-Meshes) can be found. If the networkis regular, the network looks the same from every node, hence no nodecan act as a hub. The scalability problem can be solved (or controlled,at least) by ensuring that there is a full set of two-hop routes betweenall nodes. Consider a partially connected network of N nodes, whichcontains N switching elements (which need not be the nodes themselves).If each node is connected to of order N^(1/2) switching elements, thenN^(1/2) different destinations can be reached in one hop. If theswitching element is itself connected to N^(1/2) other nodes, then allN^(3/2) nodes can be reached in two hops. The network therefore has atotal of order N^(3/2) links, far fewer than the N² links of the fullyconnected network. (If N=100, N^(3/2)=1000, N²=10000). Since all trafficis switched one extra time (a total now of three times), only 50% extraswitching capacity needs to be deployed.

Finding partially connected networks is not an easy task. Mathematicallyit consists of finding a connectivity matrix with specific properties.Let A denote a ν

b connectivity matrix that enumerates the connections between ν networknodes and b other switching elements. The component a_(ij) of matrix Ais the number of links between the i th node and the j th switchingelement. Matrix A therefore describes the number of one-hop routesbetween each pair of nodes and switching elements. The number of routeswith two hops between node i and node j is given by the number of routesbetween i and an intermediate switching element k, a_(ik), times thenumber of routes from the switching element k to node j, a_(kj), summedover all intermediate elements. We denote this matrix B and its elementsb_(kj). That is

$b_{ij} = {\sum\limits_{k}{a_{ik}{a_{kj}.}}}$

This is just the product of matrix A with itself, so that B=AA^(T). Thetwo-hop property we want to enforce, along with regularity, is:

$\begin{matrix}{B = {{A\; A^{T}} = {\begin{pmatrix}r & \lambda & \cdot & \cdot & \lambda \\\lambda & r & \; & \; & \cdot \\ \cdot & \; & \cdot & \; & \cdot \\ \cdot & \; & \; & \cdot & \cdot \\\lambda & \cdot & \cdot & \cdot & r\end{pmatrix} = {{( {r - \lambda} )I} + {\lambda\; J}}}}} & (1)\end{matrix}$where I is the ν×ν identity matrix, and J is a ν×ν matrix of ones.Equation (1) is a particular representation of a Balanced IncompleteBlock Design (BIBD), if there are exactly k ones in each column of A(i.e. each switching element is connected to exactly k nodes) [1,2].BIBD's were first used in statistical experiment design by Ronald Fisherin the 1920's. They have since found many other uses, in tournamentdesign, coding and cryptography, but not, until now, network design. Inthe combinatorics literature, BIBD's are often denoted BIBD(ν,b,r,k), orsince b and r can be determined from other parameters, as 2−(ν,kλ)designs, or as simply (ν,kλ) designs.

There are many ways of applying BIBD's in communication networks. Tomake the network more resilient, λ>1, so that there is more than onechoice of route between any two nodes. Each of these λ choices isdiverse—the routes share no common node or link—since each route musttraverse a different switching element. If one route becomesunavailable, for any reason, there is always at least one other routethat can be used.

Two Tier Applications: Stars and Areas

The most general application of BIBD's is to consider the b switchingelements as a separate switching layer. These switches do not constitutea conventional trunk or higher tier network, as they are not directlyconnected together. The terms ‘Star node’ and ‘Area node’ are used todistinguish the b switching elements that transit traffic from the νnodes that sink and source traffic. All BIBD's can be used for this typeof application, since it imposes no symmetry constraints on the BIBD. Inparticular, ν is not equal to b, and the matrix A does not have to besymmetric, i.e. a_(ij) is not equal to a_(ji).

An example is an asymmetric (7,4,2) design with ν=b. The matrix Adescribing the connectivity between area nodes 1-7 and star nodes A-G isshown in FIG. 3, together with a sketch of the network.

There are many extensions of this concept: area nodes can be generalisedto include many separate edge nodes using a common set of star nodes,etc, etc. Examples of this type of application are discussed inreference [3].

Single Tier Applications

Using BIBD's in a single tier network, the connectivity matrix A must besymmetric, and ν=b. A symmetric matrix should not be confused with asymmetric BIBD, which has a different mathematical definition. Thediagonal of A must be zero otherwise nodes would contain non-triviallinks to themselves. These constraints limit the number of suitableBIBD's. Taking a pragmatic view, one can allow many imperfections andstill produce high quality networks. In particular, if the maxim isadopted that there are at least two choices of one or two hop routesbetween every pair of nodesa _(ij) +b _(ij)>1,many good network designs that are not BIBD can be found. Of the manyclasses that have been found that fulfil this criteria, two areespecially useful (i) symmetric BIBD's with the diagonal removed (thismakes the network slightly irregular, but does not decrease networkperformance appreciably) and (ii) Strongly Regular Graphs. A StronglyRegular Graph with parameters (ν,k,λ,μ) has ν nodes without loops (i.e.links to itself—the diagonal is zero) or multiple links between nodes,and has k links to other nodes. The matrix B giving the all importanttwo hop routes isB=kI+μA+λ(J−I−A)  (2)where I is a ν×ν identity matrix, and J is a ν×ν matrix of ones, asbefore. Equation (2) says that there are λ choices of two hop routesalmost everywhere except where there are direct links, where there are μchoices of two hop routes (as well as one direct route). The amount ofextra connectivity is small, since of the ν² total two-hop routes, onlyνk of them will have more than two choices, and k is proportional toν^(1/2).

FIG. 4 shows a (9,4,1,2) Strongly Regular Graph, denoted 3² in [2]. Thestructure of the (9,4,1,2) graph, consisting of 3 groups of nodes, eachgroup consisting of a full-mesh of 3 nodes, each node being additionallyconnected to its partner in the other two groups, suggests many obviousextensions. Repeating the pattern with groups of four nodes gives theStrongly Regular Graph (16,6,2,2), which is also a (16,6,2) BIBD.Likewise the pattern can be extended ad infinitum, with 5 blocks of 5nodes, etc, all of which are Strongly Regular Graphs. Forming patternsfrom, 6 groups of 5 nodes, etc, yields graphs which are not stronglyregular, but nevertheless form excellent communications networks withdiverse mostly two choice two-hop routes.

Performance Comparisons

Since no one performance metric can characterise a network, the relativeperformance of a set of networks have been compared across a range ofmeasures. Since comparison with rings and meshes would reveal nothingnew, randomly-meshed networks with 9,16,25 and 36 nodes were compared topartially meshed networks with the same numbers of nodes [4]. Thepartially-meshed networks chosen were the 3², 4², 5² and 6² StronglyRegular Graphs from reference [2]. Randomly meshed networks weregenerated using a commercial simulated-annealing network planning toolset up to use minimum hop routing on an even traffic distribution [5].All networks were designed so that (at least) two diverse paths existfor every traffic demand. The number of links was controlled bypenalising transit traffic. With no transit traffic penalty, thenetworks become ring like; upon increasing the transit traffic penaltythe networks become more mesh-like.

When comparing any two networks, to reach any definite conclusions,assumptions must be made about traffic distribution and networkbehaviour (i.e. what sort of routing protocols are used). To make theconclusions as general as possible, it is assumed that traffic is routedusing a minimum hop scheme, with routes of equal weight being equallyutilised. This corresponds to OSPF equal-cost multi-path routing withlink weights set to one, and is representative of many commonly usedrouting schemes [6]. The traffic was assumed to be uniformlydistributed, with an arbitrary one unit of traffic demand between allnode pairs. This simplifies analysis, but is also, when designingnetworks, the least biased traffic distribution to choose in the absenceof any information about traffic distribution. The choice of trafficdistribution does not affect the primary topological design issuesconsidered.

Transit Traffic and its Distribution

Transit traffic, is a simple measure of network efficiency. For auniform traffic distribution, the transit traffic distribution isrepresentative of the route distribution across the network. FIGS. 5 aand 5 b show the transit traffic on each node and the traffic on eachlink for the randomly meshed and partially connected 9 node, 18 linknetworks respectively.

In the randomly meshed network, two nodes act as hubs, carrying far moretransit traffic than other nodes (14 and 10 units respectively).

FIG. 6 shows the mean transit traffic per node, and the sum of meantransit traffic and its standard deviation, plotted as a function of thenumber of links for all networks. The mean transit traffic is comparablein randomly meshed and partially connected networks when the numbers oflinks is equal. This shows that the randomly meshed networks are as goodas the partially meshed, using this measure of efficiency. However, thestandard deviation of the transit traffic in the randomly meshednetworks shows the traffic is unevenly distributed. For the regularnetworks, the standard deviation is zero; the traffic is perfectlydistributed. Uneven traffic distribution is not a problem for afunctioning network as individual links and nodes can be dimensionedaccordingly, but it causes problems when links or nodes fail, sincelarge amounts of transit traffic will need re-routing.

Node and Link Failure

In any network, nodes and links can always fail. The ability of anetwork to function with failed links or nodes is a vital component ofits design. Consider the failure of the busiest link (shown dotted) inthe nine node networks shown in FIG. 7. Traffic has been re-routedaround the failure and the new node and link occupancies calculated.FIG. 7 shows them expressed as a multiplier of the load in the unfailedstate.

Less spare capacity needs to be provided in the regular network to carrytraffic in the failed state, as much less traffic needs re-routing,since the transit traffic is uniformly distributed. Plotted in FIG. 8 isthe worst case increase in capacity of nodes and links that ensues whenall nodes or links are failed, one at a time.

FIG. 8 can be used to determine the worst case planning limit thatshould be used on nodes and links to ensure no network congestion infailure. Some links in the randomly meshed networks can only be run at amaximum of 30% occupancy, whereas in the regular networks, this figurevaries from 75% in the 9 node network, to 83% in the 36 node network. Inthe randomly meshed networks, planning limits must be determined forevery node and link; for the regular mesh networks, they are constantacross all nodes and links. FIG. 9 shows the total node and linkcapacity is required to support the given traffic load and survive allpossible single point of failure scenarios as a fraction of the workingnode and link capacity. Actual deployed capacity is lower for theregular mesh networks with comparable numbers of links, and even forsome networks with fewer links: making networks too sparse can often bea false economy.

Load Balancing and Uneven Traffic Distributions

Two objections might be raised to the analysis presented above: that theassumption of a uniform traffic distribution invalidates the results andthat load balancing algorithms can mitigate the effects of poor networkdesign.

The key advantage of the regular partially connected network designs isthat the one and two hop routes upon which traffic is routed (and whichmost routing algorithms would utilise first) are evenly distributedacross the network. Therefore a regular network will, almost regardlessof the route selection algorithm, spread any traffic distribution asevenly as is possible across the network. Uneven transit trafficdistribution (and the problem of hub nodes) is a function of networktopology, not traffic distribution.

Load-balancing algorithms can improve the traffic distribution in thenetwork. Using a simple load -balancing algorithm (to break ties onequal length routes, find the resource with the highest utilisation,choose the route with the lowest of the two) for both node and linksshowed that traffic could be balanced across the most highly utilisednodes or links. In effect, the two biggest hubs in the network werebalanced. This is not surprising as many equal-length alternative routestraverse both hubs: these are the nodes or links that get balanced. Forthe regular networks, since node or link occupancies are more equal,node or link—balancing algorithms tend to balance load across all nodeor links. Load-balancing algorithms can improve network balance, buttend to work better on regular networks. Algorithms that try toguarantee quality of service (QoS) would also tend to work better on aregular network. QoS algorithms typically select a short path, checkthat the required service can be supported, and route the trafficaccordingly. Only if the requested QoS cannot be guaranteed on this path(say due to resource exhaustion) will an alternative (longer) path beselected. This longer path will consume more network resources than ashorter path. If the point at which QoS algorithms choose longer pathscan be delayed, by loading traffic more evenly on the network, then thefinal capacity of the network to route traffic with a given QoS will behigher.

Network design always involves a compromise between cost andperformance; between rings and meshes. Among the near-infinite choicesbetween these extremes are mathematically perfect or near-perfectregular partial mesh networks derived from BIBD's and strongly regulargraphs. Such networks have natural traffic-balancing properties thatmake them preferable to random meshes of similar connectivity. It isadvocated using these designs for their efficiency, robustness andregularity.

Following the analysis above, communications networks based on thesuggested Balanced Incomplete Block Designs (BIBD's) and similarincidence matrices have many properties that make them especially suitedfor use in communications networks. The particular properties are:

-   -   1. All nodes are connected by routes of length maximum. 2    -   2. Multiple routes are provided that enhance load balancing and        redundancy.        Networks and Topology

The topology of an arbitrary communication network can be represented asa Graph, which is an arrangement of Nodes (or Vertices) connected byLinks (or Edges). A Node can represent a switching or routing element,or a logical aggregation of such elements. Links provide point-to-pointconnections between Nodes, and can represent physical connectivity (e.g.a fibre optic transmission system), logical connectivity (a virtualcircuit, for instance) or a logical aggregate of such.

The topology of a network can also be represented by a connectivitymatrix. If the Nodes in a network are labelled 1 . . . N, theconnectivity matrix is an ordered array of numbers with N rows and Ncolumns, with the entry in the ith row and jth column representing thenumber of Links between the ith and jth Node. We denote the entirematrix by C, and the entry in row i column j by c_(ij). An example isshown below.

$C = \begin{pmatrix}0 & 1 & 0 & 0 & 1 \\1 & 0 & 1 & 0 & 0 \\0 & 1 & 0 & 1 & 1 \\0 & 0 & 0 & 0 & 0 \\1 & 0 & 1 & 1 & 0\end{pmatrix}$

The transpose of matrix C is denoted C^(T) and is defined byinterchanging row and column indices:c^(T) _(ij)=c_(ji).

Multiplication of two matrices C and D to give E is denoted by:E=CDand defined by the following operations on the components of eachmatrix:

$e_{ij} = {{\sum\limits_{k}{c_{ik}d_{kj}}} = {\sum\limits_{k}{c_{ik}d_{jk}^{T}}}}$

A Route across the network consists of a set of Nodes and Linkstraversed in order. The Length of the Route is defined to be the numberof Links traversed. The number of Routes of Length 1 between Nodes i andj is given by the entry c_(ij) in the connectivity matrix. The number ofRoutes of Length 2 between Nodes i and j is given by the number ofRoutes from Node i to any intermediate Node k multiplied by the numberof routes from Node k to Node j. This set of numbers can be written as amatrix, which is called B.

$b_{ij} = {\sum\limits_{k}{c_{ik}c_{kj}}}$andB=CC^(T)Balanced Incomplete Block Designs

A Balanced Incomplete Block Design (BIBD) is a concept that originatesin combinatorial analysis. BIBD's solve the problem of arranging objectsinto a given number of sets under a certain set of restrictions. Aformal description, taken from “Combinatorial Theory”, Marshall Hall,(Blaisdell: Waltham Mass. 1967) is:

-   -   A balanced incomplete block design is an arrangement of ν        distinct objects into b blocks such that each block contains        exactly k distinct objects, each object occurs in exactly r        different blocks, and every pair of distinct objects a_(i),        a_(j) occurs together in exactly λ blocks.

A balanced incomplete block design can also be described by an incidencematrix. This is a matrix A with ν rows and b columns, where, if a₁ , . .. , a_(ν) are the objects and B₁ , . . . , B_(b) are the blocks, thena_(ij)=1, if a_(i)εB_(j)a_(ij)=0, if a_(i)∉B_(j)

Then, a balanced incomplete block design will have the followingproperties—

$\mspace{20mu}{{A\; A^{T}} = {\begin{pmatrix}r & \lambda & \cdot & \cdot & \cdot & \lambda \\\lambda & r & \; & \; & \; & \; \\ \cdot & \; & \cdot & \; & \; & \cdot \\ \cdot & \; & \; & \cdot & \; & \cdot \\ \cdot & \; & \; & \; & \cdot & \cdot \\\lambda & \cdot & \cdot & \cdot & \cdot & r\end{pmatrix} = {{( {r - \lambda} )I_{v}} + {\lambda\; J_{v}I_{v}}}}}$where I_(ν) is the ν times ν identity matrix, and J_(ν) is a ν times νmatrix of ones. An additional constraint is there must be exactly k onesin each column of A.Imperfect BIBD's

An Imperfect BIBD or Imperfect Network is defined as a BIBD wherein atleast one Topological Node has a missing or extra Topological Link.

BIBD's and Networks

The incidence matrix A of a block design can be used to connect some orall of the Nodes in a Network. The key property is that a particularsubset of connected Nodes have λ Routes of Length at maximum 2 betweenall distinct Nodes of that subset. The connected Nodes in this subsetwill have the following properties:

-   1. Connectivity: All Nodes are connected by λ Routes of Length 2.    (They may also be connected by Routes of Length 1, and many longer    Routes.)-   2. Balancing: If λ>1, traffic may be balanced across the A different    Routes available.-   3. Resilience: If a Node or Link on a Route fails, λ−1 equivalent    Routes can be used to carry the traffic.

It is the balancing and resilience properties, together with shortRoutes, that make these connectivity patterns so useful as networks.

SUMMARY OF THE INVENTION

According to the present invention there is provided a partiallyinterconnected topological network comprising at least six TopologicalNodes, each Topological Node having at least three point-to-pointTopological Links connecting it to some but not all of the plurality ofTopological Nodes and where there is at least one Choice of routingbetween any two Topological Nodes and where a Choice of routingcomprises either two point-to-point Topological Links connected inseries at another of the Topological Nodes or a direct point-to-pointTopological Link between the two Topological Nodes.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention will now be described by way of example, withreference to the accompanying drawings, in which:—

FIG. 1 a shows a fully-meshed network with nine nodes;

FIG. 1 b shows a ring network with nine nodes;

FIG. 2 shows a random partial mesh with nine nodes;

FIG. 3 shows a two tier application of a symmetric (7, 4, 2) design;

FIG. 4 shows a (9, 1, 4, 2) Strongly Regular Graph;

FIG. 5 a shows transit traffic for a randomly meshed 9 node, 18 linknetwork;

FIG. 5 b shows transit traffic for a partially connected 9 node, 18 linknetwork;

FIG. 6 shows the average transit traffic per node as a function of thenumber of links;

FIG. 7 a shows the change in node and link loading upon link failure fora nine node random partial mesh network;

FIG. 7 b shows the change in node and link loading upon link failure fora partially connected nine node network;

FIG. 8 shows the worst case increase in load for single failure of anynode or link;

FIG. 9 shows the increase in capacity required to survive all singlepoints of failure as a fraction of the capacity of the unfailed state;

FIG. 10 shows a connection diagram for a 16 node BIBD with single-onelink connections to 6 nodes and twin-two link connections to all nodes(16, 10, 2);

FIG. 11 shows the connection pattern (connectivity matrix) for a (16,10, 6) BIBD;

FIG. 12 shows a connection diagram for a 15 node BIBD with single-onelink connections to 8 nodes and quad-two link connections to all nodes(15, 8, 4);

FIG. 13 shows the connection pattern for a (36, 14, 6) BIBD;

FIG. 14 shows a connection diagram for a 9 node BIBD with single-onelink connections to 4 nodes and single-two link connections to the same4 nodes and twin-two link connections to the other 4 nodes;

FIG. 15 shows the connection pattern for a (25, 8, 3, 2) SRG;

FIG. 16 shows the connection pattern for a (36, 10, 4, 2) SRG;

FIG. 17 shows a connection diagram for a 10 node BIBD with single-onelink connections to 3 nodes and single-two link connections to the other6 nodes;

FIG. 18 shows the connection pattern for a (50, 7, 0, 1) SRG

FIG. 19 shows the connection pattern for a 30 node extended (25, 8, 3,2) SRG

FIG. 20 shows the connection pattern for a 35 node extended (25, 8, 3,2) SRG;

FIG. 21 shows the connection pattern for an (11, 5, 2) BIBD with atruncated diagonal;

FIG. 22 shows the connection diagram for a (19, 10, 5) BIBD beforetruncating;

FIG. 23 shows the connection diagram for a (37, 9, 2) BIBD with atruncated diagonal;

FIG. 24 shows a (16, 6, 2) BIBD (which is also a Strongly RegularGraph), forming a network of primary nodes, which has been extended byassociating two secondary nodes with each primary node and connectingeach secondary node to the same primary nodes as its associatedsecondary node;

FIG. 25 shows a (16, 6, 2) BIBD (which is also a Strongly RegularGraph), forming a network of primary nodes which has been extended byassociating two secondary nodes with each primary node and connectingeach secondary node to the same primary nodes as its associatedsecondary node;

FIG. 26 shows a (16, 6, 2) BIBD (which is also a Strongly RegularGraph), forming a network of primary nodes, which has been extended byassociating one secondary node with each primary node and connectingeach secondary node to the to the same primary nodes as its associatedsecondary node; Secondary nodes are also connected if their associatedprimary nodes are connected;

FIG. 27 shows a (16, 6, 2) BIBD (which is also a Strongly RegularGraph), forming a network of primary nodes, which has been extended byassociating two secondary nodes with each primary node and connectingeach secondary node to the same primary nodes as its associatedsecondary node; Secondary nodes are also connected if their associatedprimary nodes are connected.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

BIBD's can be used in networks in the following ways:

1. Complete Networks:

The simplest case: if the BIBD has a square, symmetric, incidence matrix(ν=b), then it can be used as the network connectivity matrix C. EveryNode in the network will have λ Routes of Length 2 to all other Nodes.It will also have Routes of Length 1 to some Nodes.

2. Two Level Networks:

Any block design can be used in a two level network, where the totalnumber of Nodes (N=ν+b) can be broken down into two categories of Node,ν of one type, b of another (the two sets of Nodes might represent Trunkand Local exchanges in a PSTN, for example), where the desirableconnectivity properties are required between the subset of ν Nodes. (Ifthe BIDB has a square symmetric incidence matrix, then A^(T)A=AA^(T),and the desired connectivity properties hold between all Nodes). The νtimes b incidence matrix A of the N times N connectivity matrix isincorporated as follows:

$C = \begin{pmatrix}0 & \cdot & \cdot & 0 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & ( {b \times b} ) & \; & \cdot & \cdot & \; & A^{T} & \; & \; & \cdot \\ \cdot & \; & \cdot & \cdot & \cdot & \; & ( {b \times v} ) & \; & \; & \cdot \\0 & \cdot & \cdot & 0 & {\cdot \;} & {\; \cdot \;} & {\cdot \;} & {\cdot \;} & {\cdot \;} & \cdot \\ \cdot & \cdot & \cdot & \cdot & 0 & \cdot & \cdot & \cdot & \cdot & 0 \\ \cdot & \; & \; & \cdot & \cdot & \cdot & \; & \; & \; & \cdot \\ \cdot & A & \; & \cdot & \cdot & \; & ( {v \times v} ) & \; & \; & \cdot \\ \cdot & ( {v \times b} ) & \; & \cdot & \cdot & \; & \; & \cdot & \; & \cdot \\ \cdot & \; & \; & \cdot & \cdot & \; & \; & \; & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & 0 & \cdot & \cdot & \cdot & \cdot & 0\end{pmatrix}$

The matrix B that gives the number of Routes of Length 2 is then

$B = \begin{pmatrix} \cdot & \cdot & \cdot & \cdot & 0 & \cdot & \cdot & \cdot & \cdot & 0 \\ \cdot & {A^{T}\; A} & \; & \cdot & \cdot & \; & \; & \; & \; & \cdot \\ \cdot & {( {b \times b} )\;} & \; & \cdot & \cdot & \; & ( {b \times v} ) & \; & \; & \cdot \\ \cdot & \cdot & \cdot & \cdot & 0 & {\mspace{11mu} \cdot} & {\cdot \;} & {\cdot \;} & {\cdot \;} & 0 \\0 & \cdot & \cdot & 0 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \; & \; & \cdot & \cdot & \; & \; & \; & \; & \cdot \\ \cdot & \; & \; & \cdot & \cdot & \; & {{A\; A^{T}}\;} & \; & \; & \cdot \\ \cdot & ( {v \times b} ) & \; & \cdot & \cdot & \; & {\;( {v \times v} )} & \; & \; & \cdot \\ \cdot & \; & \; & \cdot & \cdot & \; & \; & \; & \; & \cdot \\0 & \cdot & \cdot & 0 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \end{pmatrix}$and the desired connectivity (the AA^(T)) has been achieved between thesubset of ν Nodes as required.3. Embedded Sub-Networks:

A ν times b block design can be embedded as an arbitrary sub-network, asabove, to provide ν Nodes with the desired A Routes of Length 2 betweendifferent sub-group Nodes. If the BIDB has a square symmetric incidencematrix, then A^(T)A=AA^(T), and the desired connectivity properties holdbetween all ν+b. Nodes of the sub-network.

$C = \begin{pmatrix}\; & \; & \mspace{11mu} & \; & \; & \; & \; & \; & \; & \; & \mspace{11mu} & \mspace{11mu} & \mspace{11mu} & \; & \mspace{11mu} & \mspace{11mu} & \; \\\; & \; & \; & \; & \; & \mspace{11mu} & \; & \; & \mspace{11mu} & \; & \; & \mspace{11mu} & \; & \; & \mspace{11mu} & \; & \; \\\; & \; & \; & \mspace{11mu} & \mspace{11mu} & \; & \; & \mspace{11mu} & \; & \; & \; & \mspace{11mu} & \; & \; & \mspace{11mu} & \; & \; \\\mspace{11mu} & \mspace{14mu} & 0 & \cdot & \cdot & 0 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \mspace{11mu} & \mspace{14mu} & \; \\\; & \; & \cdot & \; & \; & \cdot & \cdot & \; & \; & A^{T} & \; & \; & \; & \cdot & \; & \; & \mspace{11mu} \\\; & \; & \cdot & {( {b \times b} )\;} & \; & \cdot & \cdot & \; & \; & ( {b \times v} ) & \; & \; & \; & \cdot & \; & \; & \; \\\; & \; & 0 & \cdot & \cdot & 0 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \; & \; & \; \\\; & \; & \cdot & \cdot & \cdot & \cdot & 0 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & 0 & \; & \mspace{11mu} & \; \\\; & \; & \cdot & \; & \; & \cdot & \cdot & \; & \; & \; & \; & \; & \; & \cdot & \; & \mspace{11mu} & \; \\\; & \; & \cdot & \; & \; & \cdot & \cdot & \; & \; & \; & \; & \; & \; & \cdot & \; & \; & \; \\\; & \; & \cdot & A & \; & \cdot & \cdot & \; & \; & \; & \; & \; & \; & \cdot & \; & \; & \; \\\; & \; & \cdot & ( {v \times b} ) & \; & \cdot & \cdot & \; & \; & {( {v \times v} )\;} & \; & \; & \; & \cdot & \; & \mspace{11mu} & \; \\\; & \; & \cdot & \; & \; & \cdot & \cdot & \; & \; & \; & \; & \; & \; & \cdot & \; & \; & \; \\\; & \; & \cdot & \; & \; & \cdot & \cdot & \; & \; & \mspace{11mu} & \; & \; & \; & \cdot & \; & \; & \; \\\; & \; & \cdot & \cdot & \cdot & \cdot & 0 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & 0 & \; & \; & \; \\\; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; \\\; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; \\\; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; \\\; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \;\end{pmatrix}$

Hence the matrix of Routes of Length 2 will be

$B = \begin{pmatrix}\; & \; & \mspace{11mu} & \; & \; & \; & \; & \; & \; & \; & \mspace{11mu} & \mspace{11mu} & \mspace{11mu} & \; & \mspace{11mu} & \mspace{11mu} & \; \\\; & \; & \; & \; & \; & \mspace{11mu} & \; & \; & \mspace{11mu} & \; & \; & \mspace{11mu} & \; & \; & \mspace{11mu} & \; & \; \\\; & \; & \; & \mspace{11mu} & \mspace{11mu} & \; & \; & \mspace{11mu} & \; & \; & \; & \mspace{11mu} & \; & \; & \mspace{11mu} & \; & \; \\\mspace{11mu} & \mspace{14mu} & \cdot & \cdot & \cdot & \cdot & 0 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & 0 & \mspace{11mu} & \mspace{14mu} & \; \\\; & \; & \cdot & {{A\mspace{14mu} A}\;} & \; & \cdot & \cdot & \; & \; & \; & \; & \; & \; & \cdot & \; & \; & \mspace{11mu} \\\; & \; & \cdot & {( {b \times b} )\;} & \; & \cdot & \cdot & \; & \; & ( {b \times v} ) & \; & \; & \; & \cdot & \; & \; & \; \\\; & \; & \cdot & \cdot & \cdot & \cdot & 0 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & 0 & \; & \; & \; \\\; & \; & 0 & \cdot & \cdot & 0 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \; & \mspace{11mu} & \; \\\; & \; & \cdot & \; & \; & \cdot & \cdot & \; & \; & \; & \; & \; & \; & \cdot & \; & \mspace{11mu} & \; \\\; & \; & \cdot & \; & \; & \cdot & \cdot & \; & \; & \; & \; & \; & \; & \cdot & \; & \; & \; \\\; & \; & \cdot & \; & \; & \cdot & \cdot & \; & \; & {{A\; A}\;} & \; & \; & \; & \cdot & \; & \; & \; \\\; & \; & \cdot & ( {v \times b} ) & \; & \cdot & \cdot & \; & \; & {( {v \times v} )\;} & \; & \; & \; & \cdot & \; & \mspace{11mu} & \; \\\; & \; & \cdot & \; & \; & \cdot & \cdot & \; & \; & \; & \; & \; & \; & \cdot & \; & \; & \; \\\; & \; & \cdot & \; & \; & \cdot & \cdot & \; & \; & \mspace{11mu} & \; & \; & \; & \cdot & \; & \; & \; \\\; & \; & 0 & \cdot & \cdot & 0 & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \; & \; & \; \\\; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; \\\; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; \\\; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; \\\; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \; & \;\end{pmatrix}$

REFERENCES

-   [1] ‘Combinatorial Theory’, M. Hall, Jr., Blaisdell (Waltham Mass.)    1967-   [2] ‘The CRC Handbook of Combinatorial Designs’, C. J. Colboum    and J. H. Dinitz (Eds), CRC Press (Boca Raton, Fla.) 1996-   [3] Patent Application No. GB 9912290.5.-   [4] NetsceneSP, a network planning and design tool which employs    simulated annealing, produced by The Network Design House, London.-   [5] ‘OSPF Version 2’, John T. Moy, RFC2328. See also OSPF, John T.    Moy, Addison Wesley Longman (Reading, Mass., USA, 1998)

1. A partially interconnected topological network, comprising: at leastsix topological nodes, wherein a topological node is a single physicalnode or a group of interconnected physical node' or part of a physicalnode or a group of interconnected physical nodes and parts of physicalnodes, each topological node having at least three point-to-pointtopological links connecting it to some, but not all, of the at leastsix topological nodes, wherein P groups of Q topological nodes arearranged so that each topological node within a group is connected toevery other topological node within that group and to exactly onetopological node in every other of the P−1 groups, where P and Q aregreat than two, and where P is not equal to Q, and where there is atleast one choice of routing between any two topological nodes, and wheresaid at least one choice of routing further comprises either twopoint-to-point topological links connected in series at another of thetopological nodes or a direct point-to-point topological link betweenthe two topological nodes.
 2. The partially interconnected topologicalnetwork as claimed in claim 1, wherein each topological node is atraffic entry point and/or a traffic exit point.
 3. The partiallyinterconnected topological network as claimed in claim 1, wherein thepartially interconnected topological network is a part of a largertopological network.
 4. The partially interconnected topological networkas claimed in claim 1, wherein each point-to-point topological linkcomprises a whole or a part of a transmission system or severaltransmission systems connected in series or in parallel.
 5. Thepartially interconnected topological network as claimed in claim 4,wherein the transmission system carries multiple circuits.
 6. Thepartially interconnected topological network as claimed in claim 1,wherein there is an equal number of point-to-point topological linksfrom each topological node.
 7. The partially interconnected topologicalnetwork as claimed in claim 1, wherein the topological network isarranged by an application of symmetric balanced incomplete blockdesigns, in which for every point-to-point topological link between twonodes A and B there is a corresponding point-to-point topological linkbetween nodes B and A, and in which no topological node is connected toitself by an external loop.
 8. The partially interconnected topologicalnetwork as claimed in claim 7, wherein there are missing topologicallinks.
 9. The partially interconnected topological network as claimed inclaim 1, wherein the topological network is arranged by an applicationof strongly regular graphs.
 10. The partially interconnected topologicalnetwork as claimed in claim 1, wherein the topological network isarranged by an application of symmetric balanced incomplete blockdesigns which for every point-to-point topological link between twonodes A and B there is a corresponding point-to-point topological linkbetween nodes B and A, and in which any external loops connecting atopological node to itself are deleted.
 11. The partially interconnectedtopological network as claimed in claim 1, wherein the network isconnected to a further similar partially interconnected topologicalnetwork having an equal or lesser number of topological nodes.
 12. Thepartially interconnected topological network as claimed in claim 1,where no topological node is connected to itself by an external loop.13. The partially interconnected topological network as claimed in claim1, wherein:(topological nodes−1)×choices=(point-to-point topological links)². 14.The partially interconnected topological network as claimed in claim 1,wherein the partially interconnected topological network is atelecommunications and/or data network.
 15. The partially interconnectedtopological network as claimed in claim 1, wherein a group of primarynodes arranged by an application of strongly regular graphs is extendedby associating one or more secondary nodes with each primary node,wherein each secondary node is connected to the same primary nodes asits associated primary node.
 16. The partially interconnectedtopological network as claimed in claim 15, wherein two secondary nodesare connected if their associated primary nodes are connected.